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At your pricepoint it's hard to nail down a definative "best" configuration as it is constantly in flux depending on availability and sales. However, I think at this time, something like this would be a good baseline to go off of:
PCPartPicker Part List
PCPartPicker Part List
CPU: AMD Ryzen 3 3100 3.6 GHz Quad-Core Processor (£93.98 @ CCL Computers)
Motherboard: Gigabyte B450M DS3H Micro ATX AM4 Motherboard (£61.99 @ CCL Computers)
Memory: Corsair Vengeance LPX 16 GB (2 x 8 GB) DDR4-3200 CL16 Memory (£58.00 @ Amazon UK)
Storage: Western Digital Blue 1 TB M.2-2280 Solid State Drive (£89.99 @ Amazon UK)
Video Card: MSI GeForce GTX 1660 6 GB VENTUS XS OC Video Card (£199.79 @ Amazon UK)
Case: Phanteks Eclipse P300A Mesh ATX Mid Tower Case (£49.99 @ Currys PC World)
Power Supply: be quiet! Pure Power 11 CM 500 W 80+ Gold Certified Semi-modular ATX Power Supply (£75.37 @ More Computers)
Total: £629.11
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2020-10-22 21:45 BST+0100
The Ryzen 3 3300X is the king of budget gaming processors at the moment, but it is having availability issues right now and spiking in price. The Ryzen 3 3100 is just marginally slower but a much better deal. Zen 3 (Ryzen 5000 series) is releasing soon, but I don't expect we'll have a sub-£100 offering soon, so the R3 3100 is still my pick for now.
16GB of RAM will cover you on any modern games, a 1TB SSD will give you enough fast storage for several large game installs, and the GTX 1660 is a good 1080p card at a sane price.
Since you're planning this as a Christmas gift though, you'll want to keep an eye on the market. We have a lot of new hardware coming out between now and then, and although it will mostly be the mid-higher range stuff, we might see a surprise, or we might see good holiday closeout deals on outgoing hardware.